Tagged Questions

2
votes
0answers
50 views

“Product” bundle notation.

Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively. Then there is an induced ...
1
vote
2answers
126 views

what is $C^{-\infty}(\mathbb{R})$

Thanks in advance. what is $C^{-\infty}(\mathbb{R})$? Is that the same as the "distribution" defined in differential geometry? It would be helpful if someone can describe it in another way ...
3
votes
2answers
249 views

Notation to work with vector-valued differential forms

What it the standard notation used while working with vector-valued differential forms? I tried using abstract index notation, for example denoting a $1$-form valued $2$-form as $P_{i[bc]}$, but I'm ...
1
vote
1answer
61 views

Notation and naming for two operations with $p$-form valued $n$-forms

While trying to answer my other question I found I never heard about vector-valued differential forms. I've been searching for them in various mathematical physics books, but didn't get too much. I'm ...
2
votes
2answers
83 views

Why parametrise a curve in this way (on the unit circle)?

I saw papers saying something like "let $\gamma:S^1 \times [0,T] \to \mathbb{R}^2$ parametrise a curve. The second interval above just makes it time dependent, but why parametrise (for fixed time) the ...
1
vote
1answer
80 views

differential and arc length notation question

Suppose $\alpha$ is a time dependent curve so that $\alpha:[0,T]\times I \to \mathbb{R} ^n$. I am a bit confused as to what the meaning of the expression $\partial_t(ds)$ is, where $ds = |\partial_x ...
6
votes
1answer
183 views

notation of derivation in differential geometry

I can't wrap my head around notation in differential geometry especially the abundant versions of derivation. Peter Petersen: Riemannian Geometry defines a lot of notation to be equal but I don't ...
1
vote
2answers
416 views

What does this symbol mean?

Below I have posted an excerpt of Lee's Book "Introduction to Smooth Manifolds" (page 371). I don't know what the symbol means that looks like a lower-right corner, and I cannot find it via the index, ...
5
votes
4answers
378 views

What does the symbol $\operatorname{Tr}$ in the Yang-Mills action mean?

I find that many authors write the Yang-Mills action as follows: $$\mathcal{J}= \int \operatorname{Tr}(F \wedge \star F).$$ I have yet to find a formal description of the symbol $\operatorname{Tr}$ ...
3
votes
1answer
237 views

Christoffel Symbol - what does a comma mean in the footer?

I am trying to understand the expression for Scalar curvature in terms of the Christoffel symbols. This is given on Wikipedia by \begin{equation} S = g^{ab}(\Gamma^c_{ab,c} - \Gamma^c_{ac,b} + ...
5
votes
1answer
209 views

idea of the star position in pullback, pushforward notation

i would like to know if there is some idea behind the position of the star in the pullback, pushforward notation or if it is just some notation without background? What's the reason for the star to ...
1
vote
1answer
198 views

The Dual Pairing

My understanding from the reading the Wikipedia article on Dual Pairs is that a dual pair is comprised of two vector spaces $X$ and $Y$ over a field $\mathbb{K}$ together with a nondegenerate ...
2
votes
0answers
119 views

bar index notation

For complex manifolds , people usually write the first fundamental form as $ds^2=g_{a\bar{b}}dz^ad\bar{z}^b$ (at least physicists) with a bar over the second index of the metric, but don't usually ...